On the equivalence of various definitions of mixed poisson processes

Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing probability distribution as well as to the mixed Poisson process in the sense of Huang is obtained, and a characterization of each one of the above mixed Poisson processes in terms of disintegrations is provided. Moreover, some examples of “canonical” probability spaces admitting counting processes satisfying the equivalence of all above statements are given. Finally, it is shown that our assumptions for the characterization of mixed Poisson processes in terms of disintegrations cannot be omitted.

Modelling Outstanding Claims with Mixed Compound Processes in Insurance

In general insurance, measuring the uncertainty of future loss payments and estimating the claims reserve are primary goals of actuaries. To deal with these tricky tasks, a broad literature is available on deterministic and stochastic approaches, most of which aims at straightforwardly modelling the overall claims reserve. In this paper by an extended, very general and reproducible case-study, we analyze the reserving process by attributing to each cell of the lower part of the run-off triangle a Compound mixed Poisson Process, calibrated upon both the numbers of claims and future average costs and considering as well the dependence among incremental claims. We provide analytically the moments of both incremental payments and the total reserve. Furthermore, we accordingly consider the probability distribution of the claims reserve, which is necessary for the assessment of the Risk Reserve capital requirement in a Solvency II framework. To test the impact of the model under different scenarios, insurers and lines of business, the case study is thoroughly analyzed by exploiting the Fisher-Lange average cost method.

Some characterizations for Markov processes as mixed renewal processes

In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector defined by Lyberopoulos and Macheras (enlarging Huang’s original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial property within this class. In concrete examples, we demonstrate how to establish the Markov property by our results. Another consequence is the invariance of the Markov property under certain changes of measures.

On large deviations for compound mixed Poisson process

This paper is designated for normal approximation to the distribution function of the compound mixed Poisson process taking into consideration large deviations both in the Cramér and power Linnik zones.

A MECHANISTIC INDIVIDUAL-BASED TWO-HOST INTERACTION MODEL FOR THE TRANSMISSION OF A PARASITIC DISEASE

A mechanistic individual-based model for the infection dynamics of the parasite Echinococcus granulosus in a two host transmission system is proposed. The model describes the individual densities of the parasites in the two host populations. The architecture consists of two sub-processes for the acquisition and severity of infection in the host populations and a superimposed infection contact scheme between the hosts. The parasite dynamics within the host population are modeled using a compound mixed Poisson process for the sheep and a shot-noise process for the dogs. All model parameters are estimated based on available data. The fitted model is then used for simulations of the transmission dynamics between the two hosts to investigate environmental factors and evaluate intervention programs.

SOFTWARE RELIABILITY MODELING BASED ON MIXED POISSON DISTRIBUTIONS

This paper considers a novel modeling framework of software reliability models (SRMs). The proposed SRMs are based on the mixed Poisson distribution (MPD), which can involve the non-homogeneous Poisson process (NHPP) based SRMs completely, but are not always equivalent to them. More precisely, the MPD-based SRMs provide a mixture of NHPPs, and their statistical properties follows the mixed Poisson process. We develop a parameter estimation method for the MPD-based SRMs based on EM algorithm. In numerical examples, we mainly investigate the difference between conventional NHPP-based SRMs and MPD-based SRMs in the viewpoints of estimating parameters and software reliability.